Inner Functions, Composition Operators, Symbolic Dynamics and Thermodynamic Formalism

TL;DR

This work develops a comprehensive thermodynamic formalism for inner functions on the unit disk, connecting composition operators, Perron–Frobenius operators, and Aleksandrov–Clark measures to obtain stochastic limit laws (CLT, LIL, ASIP) for a broad class of observables. By leveraging spectral gaps on Sobolev and Hölder spaces and by coding unit-circle dynamics via Markov partitions, the authors establish precise orbit-counting results for centered and parabolic one-component inner functions under mild integrability assumptions. The framework is extended to symbolic dynamics on countable alphabets, where Poincaré series and Tauberian arguments yield sharp asymptotics for counting preimages with bounded distortion. For one-component inner functions, a variational principle and spectral-gap results for the geometric potential are developed, yielding robust stochastic laws and enabling a detailed account of orbit growth. The parabolic case is treated through first-return maps, linking real-line expansion to symbolic dynamics and obtaining analogous counting theorems. Overall, the paper unifies operator-theoretic, probabilistic, and symbolic- dynamical methods to quantify the fine-scale statistics and orbit structure of inner function dynamics.

Abstract

In this paper, we use thermodynamic formalism to study the dynamics of inner functions $F$ acting on the unit disk. If the Denjoy-Wolff point of $F$ is in the open unit disk, then without loss of generality, we can assume that $F(0) = 0$ so that 0 is an attracting fixed point of $F$ and the Lebesgue measure on the unit circle is invariant under $F$. Utilizing the connection between composition operators, Aleksandrov-Clark measures and Perron-Frobenius operators, we develop a rudimentary thermodynamic formalism which allows us to prove the Central Limit Theorem and the Law of Iterated Logarithm for Sobolev multipliers and Hölder continuous observables. Under the more restrictive, but natural hypothesis that $F$ is a one component inner function, we develop a more complete thermodynamic formalism which is sufficient for orbit counting, assuming only the $(1+\varepsilon)$ integrability of $\log|F'|$. As one component inner functions admit countable Markov partitions of the unit circle, we may work in the abstract symbolic setting of countable alphabet subshifts of finite type. Due to the very weak hypotheses on the potential, we need to pay close attention to the regularity of the complex Perron-Frobenius operators $\mathcal L_s$ with $\text{Re }s > 1$ near the boundary. Finally, we discuss inner functions with a Denjoy-Wolff point on the unit circle. We assume a parabolic type behavior of $F$ around this point and we introduce the class of parabolic one component inner functions. By making use of the first return map, we deduce various stochastic laws and orbit counting results from the aforementioned abstract symbolic results.

Figures (1)

• Figure 1: Anatomy of a one component inner function.

Theorems & Definitions (129)

• Lemma 1.1
• Theorem 1.2
• Theorem 1.3
• Theorem 1.4
• Lemma 1.5
• Theorem 1.6: Wiener-Ikehara
• Theorem 1.7: Hardy-Littlewood
• Remark
• Theorem 1.8
• Theorem 1.9